TY - GEN

T1 - A Fractal Dimension for Measures via Persistent Homology

AU - Adams, Henry

AU - Aminian, Manuchehr

AU - Farnell, Elin

AU - Kirby, Michael

AU - Mirth, Joshua

AU - Neville, Rachel

AU - Peterson, Chris

AU - Shonkwiler, Clayton

N1 - Publisher Copyright:
© 2020, Springer Nature Switzerland AG.

PY - 2020

Y1 - 2020

N2 - We use persistent homology in order to define a family of fractal dimensions, denoted dimPHi for each homological dimension i ≥ 0, assigned to a probability measure μ on a metric space. The case of zero-dimensional homology (i = 0) relates to work by Steele (Ann Probab 16(4): 1767–1787, 1988) studying the total length of a minimal spanning tree on a random sampling of points. Indeed, if μ is supported on a compact subset of Euclidean space ℝm for m ≥ 2, then Steele’s work implies that dimPH0(μ)=m if the absolutely continuous part of μ has positive mass, and otherwise dimPH0(μ)<m. Experiments suggest that similar results may be true for higher-dimensional homology 0 < i < m, though this is an open question. Our fractal dimension is defined by considering a limit, as the number of points n goes to infinity, of the total sum of the i-dimensional persistent homology interval lengths for n random points selected from μ in an i.i.d. fashion. To some measures μ, we are able to assign a finer invariant, a curve measuring the limiting distribution of persistent homology interval lengths as the number of points goes to infinity. We prove this limiting curve exists in the case of zero-dimensional homology when μ is the uniform distribution over the unit interval, and conjecture that it exists when μ is the rescaled probability measure for a compact set in Euclidean space with positive Lebesgue measure.

AB - We use persistent homology in order to define a family of fractal dimensions, denoted dimPHi for each homological dimension i ≥ 0, assigned to a probability measure μ on a metric space. The case of zero-dimensional homology (i = 0) relates to work by Steele (Ann Probab 16(4): 1767–1787, 1988) studying the total length of a minimal spanning tree on a random sampling of points. Indeed, if μ is supported on a compact subset of Euclidean space ℝm for m ≥ 2, then Steele’s work implies that dimPH0(μ)=m if the absolutely continuous part of μ has positive mass, and otherwise dimPH0(μ)<m. Experiments suggest that similar results may be true for higher-dimensional homology 0 < i < m, though this is an open question. Our fractal dimension is defined by considering a limit, as the number of points n goes to infinity, of the total sum of the i-dimensional persistent homology interval lengths for n random points selected from μ in an i.i.d. fashion. To some measures μ, we are able to assign a finer invariant, a curve measuring the limiting distribution of persistent homology interval lengths as the number of points goes to infinity. We prove this limiting curve exists in the case of zero-dimensional homology when μ is the uniform distribution over the unit interval, and conjecture that it exists when μ is the rescaled probability measure for a compact set in Euclidean space with positive Lebesgue measure.

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U2 - 10.1007/978-3-030-43408-3_1

DO - 10.1007/978-3-030-43408-3_1

M3 - Conference contribution

AN - SCOPUS:85087744887

SN - 9783030434076

T3 - Abel Symposia

SP - 1

EP - 31

BT - Topological Data Analysis - The Abel Symposium, 2018

A2 - Baas, Nils A.

A2 - Quick, Gereon

A2 - Szymik, Markus

A2 - Thaule, Marius

A2 - Carlsson, Gunnar E.

PB - Springer

T2 - Abel Symposium, 2018

Y2 - 4 June 2018 through 8 June 2018

ER -